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Abstract.One-step collocation methods are known to be a subclass of implicit RungeKutta methods. Further, one-leg methods are special multistep one-point collocation methods. In this paper we extend both of these collocation ideas to multistep collocation methods with k previous meshpoints and m collocation points. By construction, the order is at least m + k -1. However, by choosing the collocation points in the right way, order 2m + k -1 is obtained as the maximum. There are ( mí_, ) sets of such "multistep Gaussian" collocation points.

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